# Seminars & Colloquia Calendar

## Bounds for the least solutions of quadratic inequalities

#### Thomas Hille, Yale

Location: ** Room 425**

Date & time: Tuesday, 10 March 2020 at 2:00PM - 3:00PM

Abstract: Let \(Q\) be a non-degenerate indefinite quadratic form in d variables. In the mid 80's, Margulis proved the Oppenheim conjecture, which states that if \(d \geq 3\) and \(Q\) is not proportional to a rational form then \(Q\) takes values arbitrarily close to zero at integral points. In this talk we will discuss the problem of obtaining bounds for the least integral solution of the Diophantine inequality \(|Q[x]|< \epsilon\) for any positive \(\epsilon\) if \(d \geq 5\). We will review historical, as well as recent results in this direction and show how to obtain explicit bounds that are polynomial in \(\epsilon^{-1}\), with exponents depending only on the signature of \(Q\) or if applicable, the Diophantine properties of \(Q\). This talk is based on joint work with P. Buterus, F. GĂ¶tze and G. Margulis.

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